NOV 2015: Another site that works is http://www.videograbber.net I verified that I was able to download from both Naver TVcast and Naver V. They have both an online downloader tool and an actual program you can download/install on your own computer.

JULY 2015: Another similar tool is ClipConverter.cc. I tried it, and can verify that it works well. It may also be even even better than what I recommended below, as you can choose the framerate of the video you want to download, making it perfect for those of you who demand 1080p quality. [HT to Xiao Long]

Naver runs a sort of YouTube competitor called Naver tvcast (네이버 tv캐스트), and if you've noticed that some of your favorite K-drama shows seem to have been disappearing from YouTube recently, this is likely why.

You also might have noticed that many of the usual video downloader extensions don't work on tvcast. There are some stand-alone apps that apparently can do so, including atresdownloader and Houlo Video Downloader, and the excellent youtube-dl (a nice illustrated guide to using youtube-dl to download Naver TVcast videos is here), but I bet you want something easier and something non-Windows-specific, and that will work on mobile.

Luckily, there's a simple online downloader that will do it for us. It's called "To Get FLV" and works for a variety of Korean streaming video websites. Here's a look at it in action:

Click 검색 ("Search") and in a moment a thumbnail of your video will load below the search-box.

To proceed to downloading, click the button there circled in red above, which reads 위 저작권법에 동의하며 다운로드합니다. Don't worry too much about what it says. If you've come this far, it likely won't matter to you. (OK, actually it just says that you agree that the video is for personal use only).

Step 3:

You'll arrive at a simple link page like above. Just like it says, right-click on the second link to "Save as..." and you'll download the MP4 file. I had to add the .mp4 extension manually when I tried this but otherwise it plays great.

And that's it. This convenient tool apparently works not only with tvcast, but can also download videos from a variety of other Korean sites: